Now we consider two initial states \(|\psi _{1}(0)\rangle =|e,g,0\rangle \)  and \(|\psi _{2}(0)\rangle =|g,e,0\rangle \)  (either \(a\)  or \(b\)  is initially prepared in the excited state) to compare the excitation transfers from \(a\)  to \(b\)  and from \(b\)  to \(a\) , respectively. This can be done by focusing on the dynamic evolutions of the probabilities \(P_{b,1}\)  and \(P_{a,2}\) , where \(P_{j,n}\)  (\(j=a,\,b\) ; \(n=1,\,2\) ) denotes the populations \(|c_{j}|^{2}\)  of emitter \(j\)  with initial state \(|\psi _{n}(0)\rangle \) . Moreover, we define \(\eta =d\gamma /v_{g}\)  as the separation distance between the emitters normalized by the coherence length [1]}, [2]}, [3]}. Therefore the relation between phase \(\phi \)  and the time delay reads \(\phi =\omega _{0}d/v_{g}=\omega _{0}\eta /\gamma \) . For example, if emitters \(a\)  and \(b\)  are two identical superconducting qubits (artificial atoms) with \(\omega _{0}/2\pi =3.276\textrm {GHz}\)  and \(\gamma /2\pi =29.2\textrm {MHz}\) , we have \(\phi =\lbrace 20,\,20.5,\,21\rbrace \pi \)  for \(\eta =\lbrace 0.56,\,0.574,\,0.588\rbrace \) , respectively.
